Two-Time-Scale Control of a Low-Order Nonlinear, Nonstandard System with Uncertain Dynamics
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2018 AACC. This paper extends the theory of sequential control for nonlinear, nonstandard two-time-scale systems to address uncertainties in the model structure and parameters. The approach first develops a control law based on the reduced subsystems which include the best estimates or worst cases of the uncertainties. Lyapunov design of the manifold of the fast state and the control ensures stability of the reduced subsystems. A composite Lyapunov analysis for the full-order system then yields an update law for online parameter estimation, and lower and upper bounds of the time-scale separation parameter for stability of the closed-loop system. The bounds are functions of the structural uncertainty, and reduce to the expected results in the special case of no structural uncertainty. Despite the time-derivative of the composite Lyapunov function being negative-semidefinite, convergence of the fast state to its manifold and that of the slow state to zero are proven using a result due to Barbalat's lemma. It is rigorously established that the control law can accomplish the desired objective of slow state regulation when the parameter estimation error remains bounded, but does not necessarily go to zero. The theory is demonstrated on a nonlinear, nonstandard spring-mass-damper system with uncertainties in the model of the nonlinear spring.
